Calibration of the length of a chain of single gold atoms

Calibration of the length of a chain of single gold atoms

Untiedt, C.; Yanson, A.I.; Grande, R.; Rubio-Bollinger, G.; Agraїt, N.; Vieira, S.; Ruitenbeek,

J.M. van

Citation

Untiedt, C., Yanson, A. I., Grande, R., Rubio-Bollinger, G., Agraїt, N., Vieira, S., & Ruitenbeek,

J. M. van. (2002). Calibration of the length of a chain of single gold atoms. Physical Review B,

66(8), 085418. doi:10.1103/PhysRevB.66.085418

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Calibration of the length of a chain of single gold atoms

C. Untiedt,1 A. I. Yanson,1R. Grande,2 G. Rubio-Bollinger,2 N. Agraı¨t,2S. Vieira,2and J.M. van Ruitenbeek1 1

Kamerlingh Onnes Laboratorium, Leiden University, PO Box 9504, NL-2300 RA Leiden, The Netherlands

2_{Laboratorio de Bajas Temperaturas, Departamento de Fı´sica de la Materia Condensada C-III, Universidad Auto´noma de Madrid,}

E-28049 Madrid, Spain

共Received 20 February 2002; revised manuscript received 16 May 2002; published 29 August 2002兲 Using a scanning tunnelling microscope or mechanically controllable break junction it has been shown that it is possible to control the formation of a wire made of single gold atoms. In these experiments an interatomic distance between atoms in the chain of⬃3.6 Å was reported which is not consistent with recent theoretical calculations. Here, using precise calibration procedures for both techniques, we measure the length of the atomic chains. Based on the distance between the peaks observed in the chain length histogram we find the mean value of the interatomic distance before chain rupture to be 2.5⫾0.2 Å. This value agrees with the theoretical calculations for the bond length. The discrepancy with the previous experimental measurements was due to the presence of He gas, that was used to promote the thermal contact, and which affects the value of the work function that is commonly used to calibrate distances in scanning tunnelling microscopy and mechani-cally controllable break junctions at low temperatures.

DOI: 10.1103/PhysRevB.66.085418 PACS number共s兲: 62.25.⫹g, 73.40.Jn, 73.63.Rt

I. INTRODUCTION

In the last few years, there has been a significant advance in the understanding of the electronic properties of atomic-sized contacts. This has been possible thanks to the use of two techniques: scanning tunnelling microscopy 共STM兲

共Refs. 1 and 2兲 and the mechanically controllable break

junc-tion 共MCBJ兲.3In both cases the distance between two elec-trodes is controlled by means of a piezoelectric transducer which allows for relative displacements of the electrodes down to a resolution in the range of picometers.

In these experiments the current that traverses the contact between two electrodes, at a given bias voltage, is measured as a function of the relative displacement of these electrodes. As the contact is broken, the current changes smoothly dur-ing elastic elongation stages, decreasdur-ing suddenly in plastic deformations stages.4,5 In the last stage before breaking the contact, just a few atoms determine the electronic transport and the conductance is given by the Landauer formula

G⫽2e

2

h n

兺

Tn. 共1兲

Here, N is the number of available channels for the electrons traversing the contact, e is the electron charge, h is Planck’s constant, and Tn is the transmission probability of the nth

channel. Just before the contact is broken, when there is just a single atom at the contact, the conductance for monovalent metals, such as gold, has been shown6 to be due to a single conductance channel with transmission probability close to unity and therefore with a conductance close to the value 2e2/h.

It was observed that occasionally the conductance of the one-atom contact for gold remains constant while the dis-tance between the two electrodes increases by more than an interatomic distance, as it is shown in Fig. 1. When it finally breaks, in order to make contact again it is necessary to decrease the interelectrode distance by the same distance.

From such observations it was concluded that in these cases a wire only one atom thick was formed between the two electrodes.7 These wires have also been observed by trans-mission electron microscopy 共TEM兲.8,9 The Au-Au bond length was reported to be 3.6 Å(⫾30%) and 3.6–4.0 Å in Refs. 7 and 8, respectively. It was recently shown that, apart from Au, chain formation can be observed in Pt and Ir.10,11In this paper we concentrate on Au contacts.

Several calculations have confirmed the possibility of the formation of an atomic chain of gold atoms when stretching the monatomic contact.12–16. However, in all cases a large discrepancy with the reported experimental bond length (⬃3.6 Å) was found. The calculations use different meth-ods including ab initio calculations using the local-density approximation13–15,17,18and molecular-dynamics simulations using effective-medium theory12 or tight-binding approximations.16For the wire’s equilibrium bond length the

different calculations give a distance between 2.32 and 2.55 Å, and an upper limit after stretching of 3.0 Å, much smaller than the one reported in the experiments.

In this paper we show how the interatomic distance in these atomic wires can be estimated from the conductance vs electrode displacement curves. Using this method we obtain for gold chains at low temperature 共4.2 K兲 an interatomic distance of 2.5⫾0.2 Å at the average maximum tensile stress at the moment of fracture.

II. CALIBRATION METHODS

Since the separation between the peaks in the length his-togram can provide information on the bond distance in the chains it is crucial to have a good calibration of the displace-ment of the electrodes as a function of the voltage which is applied to the piezo element of the STM or MCBJ. The various methods that we have used to calibrate our STM and MCBJ are described below.

A. Tunnel barrier

The exponential dependence of the current on the vacuum gap can be used to make a rough calibration in STM, and until very recently20it was the only way to obtain a calibra-tion of the interelectrode displacements in the MCBJ. One makes use of the well-known dependence of the tunnel cur-rent I_T between two electrodes which are separated by a distance d, when a small voltage V0 smaller than the work function of the electrodes is applied,21

IT共V0兲⫽KV0e⫺2d冑2m␾/ប

2

, 共2兲

where m is the mass of the electron, ␾ is the height of the tunnel barrier, approximately given by the mean value of the work function of the two electrodes, K is a constant which is related to the area of the electrodes and to the electronic density of states at the Fermi level.

The exponential dependence of the tunnel current with the interelectrode distance makes it very easy to control that dis-tance and this is the basis of operation of the STM. If we represent on a semilog scale the variation of the current as function of the voltage V_p applied to the piezo element共see Fig. 2兲 for the slope␥ we obtain the following expression:

␥⫽⫺

冑

P

. 共3兲

This immediately gives us a calibration of the distance as

␬⫽_⌬V⌬d

p

⫽ ⫺ប␥

2

冑

This expression is very simple and indeed in experiments with clean electrodes an exponential behavior of the current as a function of VP is found, which would make this a

suit-able method for calibration of the response of the system to the voltage applied to the piezoelectric transducer. Although a more realistic description for the tunnel barrier must in-clude electron screening effects, it has been argued22 that

these effects nearly cancel in the logarithmic derivative, at least for not too small distances.

A problem that arises when using this method is that the value of the tunnel barrier ␾is dependent on the local work function of the closest parts of the two electrodes. This local work function depends mainly on the material with some variation due to surface distortion and crystal orientation

关e.g., the work function for gold in the 共100兲 direction is 5.47

eV while for the 共111兲 direction it has a value of 5.31 eV

共Ref. 24兲兴. However, the largest deviation is due to the use of

helium. Helium gas is commonly used to promote thermal contact for cooling of the STM or MCBJ. It was generally believed that helium gas does not significantly influence the electron tunneling between two metallic electrodes. How-ever, very recently it has been found that atomic layers of adsorbed helium can affect dramatically the work function measured with this technique.23 Since the apparent work function was seen to increase for a He pressure of only 0.01 Torr by 80% above the clean surface value, errors of up to 34% are introduced in the distance calibration due to the presence of a helium atmosphere. For this reason, in the MCBJ experiments described below we avoid using helium as a thermal exchange gas.

B. Gundlach oscillations

A different method for calibrating the MCBJ, based on the Gundlach oscillations, has been developed by O.Yu. Kolesnychenko et al.20The Gundlach oscillations,25or field-emission resonances, are observed in the tunnel conductance when a voltage higher than the work function of the elec-trodes is applied between them.

As illustrated in Fig. 3 when the applied voltage V across the tunnel junction is larger than the work function of the electrodes, ␾1,2, part of the barrier region becomes classi-cally accessible. In this case the wave function of the elec-trons in the region between the electrodes will be determined by the superposition of the incoming and reflected waves at the interfaces. This mechanism will give rise to periodic maxima of the transmission as a function of bias voltage

FIG. 2. Exponential dependence of the current as function of the piezo voltage at a fixed bias voltage V⫽100 mV when the two electrodes are separated by a tunnel barrier in vacuum共a兲 or in a He atmosphere共b兲.

C. UNTIEDT et al. PHYSICAL REVIEW B 66, 085418 共2002兲

when new resonant states are formed between the electrodes. Using the model for this problem proposed by Gundlach,25Kolesnychenko et al. obtained an expression for the differential conductance as a function of bias voltage given by

dI共V兲

dV ⬃A共V兲cos关␨d共V兲兴. 共5兲

The amplitude of the oscillations, A(V), decreases with volt-age as V⫺3/2 and the argument for the cosine function is given by ␨d共V兲⫽ 4 3

冑

where F is the electric-field strength in the vacuum gap. The relation between the peak position Vn of the

oscilla-tions in Eq.共5兲 and their index can be found by equating Eq.

共6兲 to 2n␲: eVn⫽␾2⫹

冉

冑

冊

During the experiment we keep F constant by applying a feedback to the piezo voltage in order to maintain the current constant. From a plot of Vn versus n2/3the work function␾

is obtained as the intercept at the voltage axis and from the slope␴ of the curve we obtain the field strength

F⫽2

冑

3␲ប ␴

3/2_{. 共8兲}

The distance between the two electrodes will then be related to F and the applied bias voltage according to

d⫽ 1

eF共eV⫹⌬␾兲, 共9兲

where⌬␾ is the difference in the work function between the two electrodes. Using these expressions the procedure to make the calibration using the Gundlach oscillations will be as follows: we record the evolution of the conductance, as well as the piezo voltage V_p, as a function of the applied bias voltage while keeping the current constant 共see Fig. 4兲. Then using Eqs. 共7兲 and 共8兲 we can calculate the field strength F⯝1.087␴3/2 关V/nm兴. Finally using Eq. 共9兲 and the response of the feedback to the voltage changes applied to the junction we can obtain

␬⫽_⌬V⌬d

p

⫽_F1 _⌬V⌬V

p

共10兲

for the response at high voltages, where the variation is ap-proximately linear.

C. Interferometric calibration

The interferometric calibration is a very accurate method for distance calibration. We have used an all-fiber interfer-ometer similar to those used in atomic force microscopy26to calibrate our STM used in the experiments on atomic chains. A scheme of the experimental set up is shown in Fig. 5.

The tip is fastened to a z positioner which is moved by four stacks of shear piezos. To calibrate the displacement of the z positioner, the light from a laser diode is focused into a single mode optical fiber and transmitted through a 2⫻2 directional coupler which splits the beam. Part of the light is coupled to a reference photodiode which measures the inten-sity of the laser beam. This inteninten-sity is the one used as ref-erence when focusing the light. The remainder of the beam is transmitted to the end of the fiber which is placed close to a mirror glued to the rear of the z positioner. In this way an interferometric cavity is formed between the fiber end and the mirror. About 95% of the beam that reaches the fiber end is transmitted, then reflected at the mirror and directed back into the fiber, interfering with the beam reflected at the fiber

FIG. 3. Energy diagram for field-emission oscillations. Horizon-tal: z coordinate parallel to the current direction. Vertical: energy. ␾1 is the work function for the left electrode and␾2 that for the

right electrode. The chemical potential for the two electrodes are shifted by the applied voltage eV.

end. The optical path difference between both beams—twice the interferometric cavity length—makes the intensity of the resulting beam to be given by

I⫽A⫹B cos共4␲d/␭⫹␦兲, 共11兲

where d is the interferometer cavity length, A and B are constants that decrease with d, and␭ is the wavelength of the laser beam, 660 nm. The calibration is performed by ap-proaching the z positioner to the fiber until the intensity de-tected by the signal photodiode is sufficiently large. In order to vary linearly the cavity length, a voltage ramp is applied to the z direction piezos while the photocurrent of the signal photodiode is measured. Two typical calibration measure-ments, with different initial interferometer cavity lengths

共and therefore with different mean intensities兲 are shown in

the graph in Fig. 5. For a voltage span⌬Vp, the interference

pattern traces a semiperiod. From Eq.共11兲, it follows that the ratio ␬ between the displacement and the applied voltage is

␬⫽_⌬V⌬d

p⫽

␭

4⌬Vp

. 共12兲

III. EXPERIMENTAL RESULTS

For the experiments we have used gold samples of better than 99.99% purity. For the STM experiments we have cleaned the sample with an H2O-H2SO4 共1:3兲 solution and mechanically sharpened the tip, while for the MCBJ a fresh surface was formed at cryogenic vacuum when breaking the sample. The experiments were all performed at 4.2 K. The conductance curves, from which the plateau lengths are ob-tained, are all measured at a constant bias voltage of 10 mV. In Fig. 1 we show a typical experiment were an atomic chain is formed with the inset showing a histogram of last-plateau lengths. We have obtained length histograms with both STM and MCBJ.

A large number of indentation-elongation cycles of gold

nanocontacts was made. Special attention was given to in-clude a large number of atomic configurations in the statis-tics, forcing structural rearrangements of a large number of atoms with frequent deep indentations of several hundreds of nanometers between cycles.

In the case of the MCBJ we have measured for several samples plateau length histograms and each of them was calibrated by both the tunnel barrier method and by means of the Gundlach oscillations. For the tunnel barrier method we have taken a work function for gold of 5.4 eV. In this case the standard deviation in the distribution of calibration val-ues results into an error of 7%. Using this calibration we obtain for the interpeak distance in the length histogram a value of 2.5⫾0.2 Å. The calibration using the Gundlach os-cillation method was hampered most of the times by multiple tip effects in the field resonances and the response of the feedback to the applied voltage often had a very important quadratic term. Such complication appears to be characteris-tic for gold23 and the method works better for most other metals. As a result, we estimate the error in the calibration to be of the order of 20% and obtain for the interpeak distance 2.3⫾0.4 Å. We verified that the calibration obtained by the tunnel barrier method after admitting He gas into the chamber shifts the peak distance to 3.3 Å in agreement with共Ref. 7兲.

In the case of the STM configuration, the calibration has been carried out by the interferometric method. This method has the advantage of being independent of tip and sample conditions. Using different lengths of the interferometer cav-ity, a value of ␬⫽3.70⫾0.13 Å/V is obtained. The experi-ments were all performed alternating the conductance with the calibration measurements every 15 000 nanocontacts, while the instrument is maintained at 4.2 K in vacuum. In Fig. 6 we show the resultant length histogram from the STM measurements. We find here again a preference for contacts with one atom in cross section to break at specific values of length with a periodicity of 2.6⫾0.2 Å.

FIG. 5. Experimental setup for the STM configuration. The dis-placement of the tip against the sample is calibrated by an interfero-metric method which reliability and accuracy remain unaffected by the environment. The inset shows two different traces of the inter-ference pattern measured with different initial interferometer cavity lengths.

FIG. 6. Histogram of lengths for the last conductance plateau obtained in 65 000 indentations made with the STM. We define the length of the last plateau as the distance between the point where the conductance drops below 1.2 times the conductance quantum and the one where the contact breaks.

C. UNTIEDT et al. PHYSICAL REVIEW B 66, 085418 共2002兲

IV. DISCUSSION

The linear bond between two gold atoms is up to three times stronger than a bulk bond, as found in experiments and simulations.11,19,27A single-atom gold contact can sustain a maximum tensile force of 1.5 nN and before this limit is reached it is likely that atoms are pulled out of their position in the banks on either side of the contact. By repeating such atomic structural changes in the banks the one-atom contact evolves into a chain several atoms long. The chain finally breaks when the tensile force necessary to incorporate an-other atom from the nearby electrodes into the chain is higher than the breaking force of the chain itself. There will be characteristic interelectrode distances for which a chain of

n atoms is likely to break, as we will argue next.

Let us first discuss a length histogram for metals that do not easily form atomic chains, such as the 4d metals Rh, Pd, and Ag investigated in Ref. 10. For these metals the length histogram shows only a single peak, usually at a shorter length than the first peak in the length histogram for Au. We start counting the length of the plateau when the conductance drops to a typical value for a single atom contact, e.g., below 1.2 G0 for Ag. When pulling further the conductance re-mains roughly at this value while the bonds of the atom with the banks and those inside the banks are being stretched. As soon as the stored elastic energy reaches a maximum the contact breaks. The breaking point depends on the local atomic configurations in the banks near the contact and this leads to a certain width in the peak distribution. Thermal activation over the breaking barrier will also lead to a statis-tical distribution of observed values. The peak position in the length histogram shows the most probable length over which a one-atom contact can be stretched.

For chain-forming metals such as Au the first peak in the length histogram has the same interpretation as for those that break at a one-atom contact. Its position is at a longer length reflecting the stronger bond for low-coordination Au atoms. For all configurations giving rise to the distribution under the first peak there are equivalent configurations with the central atom replaced by two, three, or more atoms, forming a chain. These will give rise to additional peaks in the length histo-gram at multiples of the Au-Au bond distance in the chains, but stretched close to the breaking point. These distances are the ones at which the structure reaches the maximum tensile stress while it is not possible to introduce a new atom into the chain to relax it. If we consider that the force needed to break an atomic chain, Fb, is independent of the length of

the chain28 then the interelectrode distance at which the

n-atoms chain breaks can be written as

L_n⫽nL_at⫺at⫹共n⫹1兲Fb Ka

, 共13兲

where Lat⫺at is the interatomic distance when no tension is

applied and Ka the elastic constant of the bond between

at-oms in the chain. Therefore the distance between the peaks in the plateau length histogram will be constant and equal to

⌬⫽Lat⫺at⫹Fb/Ka, or in other words, equal to the

inter-atomic distance stretched to the point of breaking.

In this argumentation we have assumed that the banks are not shortened between the point at which the conductance first is seen to drop to the one-atom level and the final break-ing point. As long as we limit the discussion to chains of only a few atoms in length this will be correct since the number of atomic layers in the banks will not be modified. Note that our value for the bond distance is based on the first two to four peaks and that atoms may fold in from both sides. Those events that result in a significant modification in the structure and effective length of the banks will only con-tribute to a smooth background in the length histogram. Only the chain-forming processes that conserve the structure of the banks are expected to be responsible for peaks at regular spacing in the length histogram, and these are thus expected to correspond to the atom-atom distance in the chains. The Au-Au distance is measured from the distance between the peaks in the histogram, and we remark that the position of the first peak 共relative to zero length兲 can differ from this value. For Au the first peak is nearly equal to the distance between the peaks, but different values have indeed been obtained, e.g., in the case of Pt chain length histograms.10

The bond distance near the anchoring points of the chain to the banks are expected to be about 10% shorter than the bond distance in the middle of the chain, as illustrated in the calculations by da Silva et al.16A small variation in the bond length is consistent with our data, as can bee seen from the position of the fourth peak in Fig. 1. The fact that we derive our values for the bond distance mainly from the first three peaks implies that our result is biased toward the smaller distances at the anchoring points.

V. CONCLUSIONS

We have applied different calibration techniques for the MCBJ and STM in order to obtain a more accurate value for the distance between peaks in length histograms of the last plateau of conductance before rupture of gold contacts at low temperature. The values obtained for the interatomic distance in a chain of gold atoms at the point of breaking are 2.5

⫾0.2, 2.3⫾0.4, and 2.6⫾0.2 Å. We obtain an overall value

for the interatomic distance of 2.5⫾0.2 Å, which closely agrees with results from model calculations.

ACKNOWLEDGMENTS

This work is part of the research program of the ‘‘Stich-ting FOM,’’ which is financially supported by NWO, it was funded by the DGI under Contract No. MAT2001-1281 and the research has been supported by the Marie Curie Program of the European Community under Contract No. HPMF-CT-2000-0072. R.G. acknowledges financial support from U.A.M. We thank O.I. Shklyarevskii and R.H.M. Smit for stimulating discussions.

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